[meteorite-list] What If Copernicus Was Wrong?

From: JoshuaTreeMuseum <joshuatreemuseum_at_meteoritecentral.com>
Date: Thu, 10 Sep 2009 15:55:31 -0400
Message-ID: <87089717A5464648A4BF96994FEC7FCB_at_ET>

Dark Energy v. The Void: What if

Copernicus was Wrong?

Living in a Void: Testing the Copernican Principle with Distant Supernovae

Timothy Clifton,? Pedro G. Ferreira, and Kate Land

Oxford Astrophysics, Physics, DWB, Keble Road, Oxford, OX1 3RH, UK

A fundamental presupposition of modern cosmology is the Copernican
Principle; that we are not

in a central, or otherwise special region of the Universe. Studies of Type
Ia supernovae, together

with the Copernican Principle, have led to the inference that the Universe
is accelerating in its

expansion. The usual explanation for this is that there must exist a 'Dark
Energy', to drive the

acceleration. Alternatively, it could be the case that the Copernican
Principle is invalid, and that

the data has been interpreted within an inappropriate theoretical
frame-work. If we were to live in

a special place in the Universe, near the centre of a void where the local
matter density is low, then

the supernovae observations could be accounted for without the addition of
dark energy. We show

that the local redshift dependence of the luminosity distance can be used as
a clear discriminant

between these two paradigms. Future surveys of Type Ia supernovae that focus
on a redshift range

of  0.1 ? 0.4 will be ideally suited to test this hypothesis, and hence to
observationally determine

the validity of the Copernican Principle on new scales, as well as probing
the degree to which dark

energy must be considered a necessary ingredient in the Universe.

The concordance model of the Universe combines two

fundamental assumptions. The first is that space-time

is dynamical, obeying Einstein's Equations. The second

is the 'Cosmological Principle', that the Universe is then

homogeneous and isotropic on large scales - a generalisation

of the Copernican Principle that "the Earth is

not in a central, specially favored position" [1]. As a result

of these two assumptions we can use the Freidmann-

Robertson-Walker (FRW) metric to describe the geometry

of the Universe in terms of a single function, the scale

factor a(t), which obeys

H2 =

8G

3

 ?

k

a2 (1)

where H ? ? a/a is the Hubble rate,  is the energy density,

k is the (constant) curvature of space, and overdots

denote time derivatives. The scale factor can then be

determined by observing the 'luminosity distance' of astrophysical

objects. At small z ? a0/a(t)?1 this is given

by

H0DL ? cz +

1

2

(1 ? q0)cz2, (2)

where q ? ??aa/ ?a2 is the deceleration rate, and subscript

0 denotes the value of a quantity today. Recent measurements

of (z, DL) using high redshift, Type Ia Supernovae

(SNe) have indicated that q0 < 0, i.e. the Universe

is accelerating in its expansion [2, 3]. Accelerating expansion

is possible in an FRW universe if a fraction of

 is in the form of a smoothly distributed and gravitationally

repulsive exotic substance, often referred to as

Dark Energy [4]. The existence of such an unusual substance

is unexpected, and requires previously unimagined

amounts of fine-tuning in order to reproduce the observations.

Nonetheless, dark energy has been incorporated

into the standard cosmological model, known as CDM.

Electronic address: tclifton at astro.ox.ac.uk

An alternative to admitting the existence of dark energy

is to review the postulates that necessitate its introduction.

In particular, it has been proposed that the SNe

observations could be accounted for without dark energy

if our local environment were emptier than the surrounding

Universe, i.e. if we were to live in a void [5, 6, 7].

This explanation for the apparent acceleration does not

invoke any exotic substances, extra dimensions, or modifications

to gravity - but it does require a rejection of the

Copernican Principle. We would be required to live near

the centre of a spherically symmetric under-density, on

a scale of the same order of magnitude as the observable

Universe. Such a situation would have profound consequences

for the interpretation of all cosmological observations,

and would ultimately mean that we could not

infer the properties of the Universe at large from what

we observe locally.

Within the standard inflationary cosmological model

the probability of large, deep voids occurring is extremely

small. However, it can be argued that the centre of a

large underdensity is the most likely place for observers

to find themselves [8]. In this case, finding ourselves in

the centre of a giant void would violate the Copernican

principle, that we are not in a special place, but it may

not violate the Principle ofMediocrity, that we are a 'typical'

set of observers. Regardless of what we consider the

a priori likelihood of such structures to be, we find that

it should be possible for observers at their centre to be

able to observationally distinguish themselves from their

counterparts in FRW universes. Living in a void leads to

a distinctive observational signature that, while broadly

similar to CDM, differs qualitatively in its details. This

gives us a simple test of a fundamental principle of modern

cosmology, as well as allowing us to subject a possible

explanation for the observed acceleration to experimental

scrutiny.

Some efforts have gone into identifying the observational

signatures that could result from living in a void.

The cosmic microwave background (CMB) supplies us

with the tight constraint that we must be within 15 Mpc

of the center of the void [9]. There have also been some

attempts at calculating predictions for CMB anisotropies

and large scale-structure [10, 11, 12], as well as the kinematic

Sunyaev-Zeldovich effect [13].

General Relativity allows a simple description of timedependent,

spherical symmetric universes: the Lema^itre-

Tolman-Bondi (LTB) models [14, 15, 16], whose lineelement

is

ds2 = ?dt2 +

a2

2(t, r)dr2

1 ? k(r)r2 + a2

1(t, r)r2d2, (3)

where a2 = (ra1)', and primes denote r derivatives . The

old FRW scale factor, a, has now been replaced by two

new scale factors, a1 and a2, describing expansion in the

directions tangential and normal to the surfaces of spherical

symmetry. These new scale factors are functions of

time, t, and distance, r, from the centre of symmetry,

and obey a generalization of the usual Friedman equation,

(1), such that

 ? a1

a12

=

8G

3

~ ?

k(r)

a2

1

. (4)

Here ~ = m(r)/a3

1, and is related to the physical energy

density by  = ~+~'ra1/3a2. The two free functions, k(r)

and m(r), correspond to the curvature of space, and the

distribution of gravitating mass in that space. We choose

initial conditions such that the curvature is asymptotically

flat with a negative perturbation near the origin,

and so that the gravitational mass is evenly distributed.

As the space-time evolves the energy density in the vicinity

of the curvature perturbation is then dispersed, and

a void forms. Observations of distant astro-physical objects

in this space-time obey a distance-redshift relation

DL = (1 + z)2rEa1(tE, rE) (5)

where

1 + z = expZ rE

0

( ?a1r)'

?1 ? kr2

dr, (6)

and subscript E denotes the value of a quantity at the

moment the observed photon was emitted. This expression

is modified from equation (2), allowing for the possibility

of apparent acceleration without dark energy.

We find that the form of the void's curvature profile

is of great importance for the observations made by astronomers

at its centre. In Figure 1 we plot some simple

curvature profiles, together with the corresponding distance

moduli as functions of redshift (distance modulus,


dm, is defined as the observable magnitude of an astrophysical

object, minus the magnitude such an object

would have at the same redshift in an empty, homogeneous

Milne universe). It is clear from Figure 1 that

for the void models there is a strong correlation between

k(r) and
dm; at low redshifts
dm(z) traces the shape

of k(r). Hence, for a generic, smooth void
dm starts

off with near zero slope, where it is locally very similar

to a Milne universe, it then increases at intermediate z,

and later drops off like an Einstein-de Sitter universe.

For CDM, we have
dm ? ?5

2 q0z at low z, i.e. a

http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.1443v2.pdf



Phil Whitmer
Received on Thu 10 Sep 2009 03:55:31 PM PDT